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I see the image contains the following factoring problems that you would like to solve:

**52)** \( x^{\frac{7}{3}} - x^{\frac{5}{3}} - 2x^{\frac{1}{3}} \)

**53)** \( 8x^{\frac{4}{3}} + 4x^{\frac{5}{3}} \)

**54)** \( 7x^{\frac{3}{7}} - 14x^{\frac{6}{7}} + 21x^{\frac{10}{7}} \)

**55)** \( 7x^{\frac{-1}{3}} + 70x \)

Let me now go ahead and solve them by factoring each expression completely.

### Problem 52: \( x^{\frac{7}{3}} - x^{\frac{5}{3}} - 2x^{\frac{1}{3}} \)
We can factor out the smallest power of \( x \), which is \( x^{\frac{1}{3}} \):
\[
x^{\frac{1}{3}} \left( x^2 - x - 2 \right)
\]
Now, factor the quadratic expression:
\[
x^2 - x - 2 = (x - 2)(x + 1)
\]
Thus, the fully factored form is:
\[
x^{\frac{1}{3}} (x - 2)(x + 1)
\]

---

### Problem 53: \( 8x^{\frac{4}{3}} + 4x^{\frac{5}{3}} \)
We can factor out the greatest common factor, which is \( 4x^{\frac{4}{3}} \):
\[
4x^{\frac{4}{3}} (2 + x)
\]
Thus, the expression is completely factored.

---

### Problem 54: \( 7x^{\frac{3}{7}} - 14x^{\frac{6}{7}} + 21x^{\frac{10}{7}} \)
Factor out the greatest common factor, which is \( 7x^{\frac{3}{7}} \):
\[
7x^{\frac{3}{7}} \left( 1 - 2x^{\frac{3}{7}} + 3x^{\frac{7}{7}} \right)
\]
Since \( x^{\frac{7}{7}} = x \), we can rewrite the expression:
\[
7x^{\frac{3}{7}} \left( 1 - 2x^{\frac{3}{7}} + 3x \right)
\]
This is the completely factored form.

---

### Problem 55: \( 7x^{\frac{-1}{3}} + 70x \)
Factor out the greatest common factor, which is \( 7x^{\frac{-1}{3}} \):
\[
7x^{\frac{-1}{3}} \left( 1 + 10x^{\frac{4}{3}} \right)
\]
This is the fully factored form.

---

Would you like any further clarifications on the steps or the results? 

Here are some related questions:
1. How do you determine the greatest common factor for algebraic expressions?
2. What is the process to factor quadratics like \( x^2 - x - 2 \)?
3. How can exponents be simplified when factoring expressions with fractional powers?
4. What are common techniques for factoring cubic expressions?
5. How do you handle negative exponents in factored expressions?

**Tip:** Always start factoring by finding the greatest common factor (GCF) before moving on to more complex methods like factoring quadratics or grouping.

Factor each expression completely:
1) x^(7/3) - x^(5/3) - 2x^(1/3)
2) 8x^(4/3) + 4x^(5/3)
3) 7x^(3/7) - 14x^(6/7) + 21x^(10/7)
4) 7x^(-1/3) + 70x

Learn how to factor algebraic expressions with fractional exponents, using the greatest common factor method. This problem covers algebraic factoring techniques suitable for students in Grades 9-12.

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